The PCs are currently at Stormgrove Manor near the Lilal Gate in the southeast part of the district. (Reminder that North is towards the left side of the map (your left) where the Gardens are located.) The PCs can get to both the Black Cat Tavern and the Jade Thicket through the Green Gateway, which leads into South Ward. Also in that ward is Spindle's Apothocary.
To get to the gardens at Hollingsworth Estate, the PCs would have to go through Patrons' Gate and Masque Arch, They would be back near the Dragori Embassy and the Gate of Scales. The garden is at the back of the estate, which is protected by a encircling wall. It is still unknown how an assassin could have gotten in to the estate to kill the Lord Mayor.
If asked, Florrie will tell the PCs that there aren't any tunnels of the Lantern Depths that go to the Lord Mayor's estate. Only the Pomander Estate has a tunnel that connects to the Lantern Depths, as she is the organizations patron among The Five. She helped create the Lanterns. While not every ward in Old City has access to a tunnel, there are tunnels (that she knows of) that come up in the Museum Ward, Theater Ward, Traders' Ward (south of South Ward), and Ward of Soldiers (where the barracks are located but not into the barracks).
To get to the Crossed Candles (the suspected foreign thieves guild), the PCs would have to go back to the Quill Gate and pass through The Trench and then down through Blade Gate into the Golden Ward.